PHYSICAL REVIEW B 99, 235125 (2019)

Symmetry representation approach to topological invariants in C2zT -symmetric systems

Junyeong Ahn and Bohm-Jung Yang*

Department of Physics and Astronomy, Seoul National University, Seoul 08826, Korea;Center for Correlated Electron Systems, Institute for Basic Science (IBS), Seoul 08826, Korea;

and Center for Theoretical Physics (CTP), Seoul National University, Seoul 08826, Korea

(Received 26 October 2018; published 11 June 2019)

We study the homotopy classification of symmetry representations to describe the bulk topological invariantsprotected by the combined operation of a twofold rotation C2z and time-reversal T symmetries. We definetopological invariants as obstructions to having smooth Bloch wave functions compatible with a momentum-independent symmetry representation. When the Bloch wave functions are required to be smooth, the informationon the band topology is contained in the symmetry representation. This implies that the d-dimensional homotopyclass of the unitary matrix representation of the symmetry operator corresponds to the d-dimensional topologicalinvariants. Here, we prove that the second Stiefel-Whitney number, a two-dimensional (2D) topologicalinvariant protected by C2zT , is the homotopy invariant that characterizes the second homotopy class of thematrix representation of C2zT . As an application of our result, we show that the three-dimensional (3D) bulktopological invariant for the C2zT -protected topological crystalline insulator proposed by C. Fang and L. Fuin Phys. Rev. B 91, 161105 (2015), which we call the 3D strong Stiefel-Whitney insulator, is identical tothe quantized magnetoelectric polarizability. The bulk-boundary correspondence associated with the quantizedmagnetoelectric polarizability shows that the 3D strong Stiefel-Whitney insulator has chiral hinge states aswell as 2D massless surface Dirac fermions. This shows that the 3D strong Stiefel-Whitney insulator hasthe characteristics of both the first- and the second-order topological insulators, simultaneously, which isin consistence with the recent classification of higher-order topological insulators protected by an order-twosymmetry.

DOI: 10.1103/PhysRevB.99.235125

I. INTRODUCTION

Topological crystalline insulators (TCIs) are insulatorswhose bulk properties cannot be adiabatically connected tothose of atomic insulators due to symmetry [1–4]. Accord-ingly, the topological invariant of a TCI can generally beconsidered as an obstruction to finding exponentially local-ized symmetric Wannier functions. Since the construction ofsymmetric Wannier functions requires both the exponentiallocalization of the wave functions and the invariance undersymmetry, the topological invariant of a TCI can be definedin two different ways. First, when the symmetry represen-tation is trivial in momentum space, meaning that it can beinduced by a symmetry representation of Wannier functionsin real space, the topological invariant of a TCI becomes anobstruction to having smooth wave functions in momentumspace.1 On the other hand, when the wave functions areassumed to be smooth in the Brillouin zone, the topologicalinvariant of a TCI is encoded in the matrix representation

*bjyang@snu.ac.kr1Topological obstruction to smoothness is due to the obstruction

to continuity because topology is about continuity. In practice, theobstruction to smoothness is identical to the obstruction to continuitybecause the Bloch wave functions are smooth in topologically trivialphases. Let us note that a nontrivial band topology does not induceBloch wave functions that are continuous but are not smooth.

of the symmetry operator, the so-called sewing matrix, andappears as an obstruction to finding a trivial sewing matrixin the Brillouin zone. Although both approaches eventuallylead to the same classification of TCIs, the different ways ofdefining a topological invariant provide complementary viewsof understanding the nature of the relevant TCIs. Identifyingthe topological obstruction under symmetry constraints andthe nature of the associated topological invariants, taking intoaccount all possible space groups and magnetic space groupsin crystals, is definitely one central issue in the study of TCIs.

Recently, it has been shown that a two-dimensional (2D)system with a magnetic symmetry under C2zT can be charac-terized by the first Stiefel-Whitney number w1 and the secondStiefel-Whitney number w2 [5,6]. As C2zT is a local sym-metry operator in momentum space, and also an antiunitarysymmetry operator that satisfies (C2zT )2 = 1, its constant rep-resentation can be taken to be the identity matrix: C2zT |unk〉 =|unk〉 [7]. This gauge choice is called a real gauge because theresidual gauge transformation belongs to a real unitary group,i.e., an orthogonal group. As the symmetry representation isconstant in this gauge, topological invariants are defined asobstructions to the smoothness of Bloch wave functions. Therelevant one-dimensional (1D) and 2D topological invariantsare w1 and w2. w1 is equivalent to the quantized Berryphase, and w2 is identical to the Z2 monopole charge [8–10]characterizing a nodal line semimetal in PT -symmetric three-dimensional (3D) spinless fermion systems [6]. In terms ofmore physical concepts, w1 is the quantized electric dipole

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JUNYEONG AHN AND BOHM-JUNG YANG PHYSICAL REVIEW B 99, 235125 (2019)

moment, and w2 is the quantized electric quadrupole moment[6]. In crystals, the electric quadrupole moment is not welldefined when the electric dipole moment is nontrivial [11].Similarly, w2 becomes a well-defined 2D topological invariantof an insulator only when w1 is trivial [6], and the insulatorwith w2 = 1 was dubbed a 2D Stiefel-Whitney insulator(SWI) [6].

The 2D SWI is a fragile topological insulator [12], whichmeans that its Wannier obstruction is fragile against addingtrivial bands (i.e., bands with w1 = w2 = 0): it has a Wannierobstruction when the number of occupied bands is two, butthe obstruction disappears when an additional trivial band isadded. After the Wannier obstruction disappears, the resultingatomic insulator with w2 = 1 still carries a nonzero electricquadrupole moment originating from the Wannier centersresiding on the boundary of the unit cell, so it correspondsto an obstructed atomic insulator [13–17]. The fragile na-ture of the 2D SWI is also reflected in the Whitney sumformula for Stiefel-Whitney numbers, which is an algebraicrule for adding w1 and w2 of the subbands below the Fermilevel [17].

In this paper, we revisit the C2zT -protected topologicalinvariants in the perspective of the homotopy classification ofsymmetry representations in momentum space [18–21]. Whenwe choose a smooth gauge instead of a real gauge, a non-trivial topology should manifest as an obstruction to taking aconstant symmetry representation. Accordingly, in principle,a homotopy classification of the corresponding sewing matrixshould give a classification of topological phases [20]. Here,we establish the relationship between the representation ofC2zT symmetry and the second Stiefel-Whitney number. Tothis end, we define momentum-independent symmetry repre-sentations as a trivial symmetry representation. This definitionallows a finer classification than the one based on the Wannierobstruction because even obstructed atomic insulators, whichare all trivial in terms of the Wannier obstruction, can bedistinguished in this way. In real space, trivial insulators in ourdefinition correspond to the atomic insulators whose Wanniercenters can be adiabatically deformed to the center of the unitcell. Although this definition of topological triviality dependson the choice of the unit cell, it is useful because it provdiesa relative classification of the obstructed atomic insulators.Given this definition, we can use the usual homotopy group toclassify topological insulators because a constant map formsthe trivial element of the homotopy group. Explicitly, we findthat the dth homotopy class of the sewing matrix directlygives the conventional d-dimensional topological invariants,that is, the Stiefel-Whitney numbers w1 and w2 for d = 1and 2, and the magnetoelectric polarizability P3 for d = 3.Let us note that although the relation between the sewingmatrices and topological invariants has already been shownfor w1 and P3 by using the Berry connection and curvature,such a relation for w2 has not been known yet. A merit of thisapproach based on the homotopy class of the sewing matrix isthat it provides the Whitney sum formula for Stiefel-Whitneynumbers in a more comprehensive way as compared to theformulation in a real gauge as shown below.

As an application of our approach, we derive the relationbetween the second Stiefel-Whitney number and the higher-order band topology of the three-dimensional topological

FIG. 1. 3D strong Stiefel-Whitney insulator (SWI) protected byC2zT symmetry. (a) Schematic figure describing the second Stiefel-Whitney numbers on the C2zT -invariant planes in momentum space.In a 3D strong SWI, w2(kz = π ) − w(kz = 0) = 1 modulo two.(b) Schematic figure describing the gapless states on the surface andhinges in real space. An odd number of 2D Dirac fermions appear oneach of the top and bottom surfaces. 1D chiral fermions appear onthe side hinges.

insulator with C2zT symmetry proposed in Ref. [7]. In 3D mo-mentum space, there are two C2zT -invariant planes with kz =0 and kz = π , and the corresponding second Stiefel-Whitneyinvariants can be written as w2(0) and w2(π ), respectively.Thus, a 3D Z2 topological invariant �3D can be defined as�3D ≡ w2(π ) − w2(0) [7]. Since �3D originates from w2 inC2zT -invariant planes, we call the 3D topological insulatorwith �3D = 1 as a 3D strong Stiefel-Whitney insulator. The3D strong SWI was originally proposed to have only Diracsurface states as anomalous boundary states on the two C2zT -invariant surfaces, which are normal to the z direction [7].However, recent classifications [22,23] of higher-order topo-logical insulators [11,15,22–45] indicate that the 3D strongSWI has additional anomalous chiral states on side hinges, so-called chiral hinge states (Fig. 1). That is, the 3D strong SWIis a mixed-order topological insulator because the nth-ordertopological insulator is defined by the presence of (d − n)-dimensional anomalous boundary states. Below we show thatthe mixed-order band topology in this system results fromthe nontrivial magnetoelectric polarizability P3 of the bulk,which satisfies �3D = 2P3. This relation is derived based onthe fact that the 3D topological invariant corresponds to thethird homotopy class of the sewing matrix.

This paper is organized as follows. In Sec. II, we ana-lyze the general properties of the sewing matrix for C2zTsymmetry. Our main technical tool is the exact sequences ofhomotopy groups that have been used to classify the spaceof Hamiltonians for topological insulators [46–48]. Using theexact sequences, in one and two dimensions, we show that thehomotopy class of the sewing matrix gives the same classifica-tion of topological phases as the homotopy class of the Hamil-tonian spaces. Then, the connection between the second andthird homotopy classes of the sewing matrix is shown. Thesegeneral results are elaborated further in the following sections.In Secs. III and IV, we show the explicit relations between thesewing matrix in a smooth gauge and the transition functionsin a real gauge. The first and second Stiefel-Whitney numbersdefined in a real gauge are matched to the first and secondhomotopy classes of the sewing matrix in a smooth gauge.In 3D, we first determine the condition for �3D to be a well-defined topological invariant in Sec. V. After that, we showthat the bulk magnetoelectric polarizability is determined bythe second Stiefel-Whitney numbers of C2zT -invariant planes

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when �3D is well defined in Sec. VI. We review the bulk-boundary correspondence between the anomalous boundarystates and the bulk magnetoelectric polarizability in Sec. VII,and demonstrate, by using a tight-binding toy model, that the3D topological insulator with �3D = 1 has both anomalousDirac surface states and chiral hinge states in Sec. VIII.Finally, we discuss some generalizations of our results inSec. IX.

II. HOMOTOPY GROUPS OF THE SEWING MATRIX

Let us begin by studying general aspects of the homotopygroups of the sewing matrix G for C2zT . G is defined as

Gmn(k) = ⟨um(−C2zk)

∣∣C2zT |unk〉 , (1)

where −C2zk = (kx, ky,−kz ) and |unk〉 is the cell-periodic partof the Bloch state. We are interested in the ground state ofthe system and study the topology of the occupied states,so hereafter we assume that m and n run over the indicesof occupied bands. Since (C2zT )2 = (C2z )2T 2 = 1 in bothspinless and spinful systems, G satisfies

Gmn(k) = Gnm(−C2zk). (2)

Under a gauge transformation |unk〉 → |u′nk〉 = Umn(k)|umk〉,

the sewing matrix transforms as

Gmn(k) → G′mn(k) = [U †(−C2zk)G(k)U ∗(k)]mn, (3)

where G′mn(k) = 〈u′

m(−C2zk)|C2zT |u′nk〉. If we choose smooth

wave functions for occupied states, the corresponding sewingmatrix also becomes smooth. The nontrivial homotopy classof G characterizes the obstruction to taking a uniform repre-sentation G(k) = G0 independent of k.

On a C2zT -invariant plane, either the kz = 0 or π plane,GT (k) = G(k) according to Eq. (2). Such a symmetric unitarymatrix can be written as

G(k) = UG(k)U TG (k), (4)

where UG is a unitary matrix describing a unitary transforma-tion from a smooth gauge to a real gauge. As a redefinitionUG(k) → O(k)UG(k) for any O(k) ∈ O(N ) does not changeG(k), we obtain

G(k) ∈ U (N )/O(N ), (5)

on C2zT -invariant planes, where N denotes the number ofoccupied bands.

Since a nontrivial homotopy class of G(k) is an obstructionto taking a constant symmetry representation, it classifiespossible topological phases for N occupied bands. To geta well-defined classification of topological phases, however,one should carefully identify the homotopy classes that arerelated to each other by gauge transformations, because asmooth gauge transformation can change the homotopy classof G. After that, the homotopy classification of the sewingmatrix should be consistent with the classification of theHamiltonian space. In fact, we show below that

πd [U (N )/O(N )]

GaugeDOF πd

[O(N + M )

O(N ) × O(M )

]M→∞

, (6)

FIG. 2. Effective domain for the sewing matrix. (a) A planerepresenting the 3D Brillouin zone. The yellow region shows theeffective Brillouin zone, and the red region with kz = 0 is a C2zT -invariant plane. The kz = π plane is assumed to be topologicallytrivial. (b) A 3-sphere equivalent to the 3D Brillouin zone. (c) Thesewing matrix G in the yellow region and on its boundary (red).

where d = 1, 2,2 GaugeDOF is the image of the map j∗ :πd [U (N )] → πd [U (N )/SO(N )] that is induced by the pro-jection j : U (N ) → U (N )/O(N ), and O(N + M )/[O(N ) ×O(M )] is the classifying space of the real (i.e., C2zT -symmetric) Hamiltonians for N occupied and M unoccupiedbands [49].

The above equivalence can be explicitly shown in twosteps. First, we use that

πd

[O(N + M )

O(N ) × O(M )

]M→∞

πd−1[O(N )], (7)

which states that the d-dimensional topological phase de-scribed by a real Hamiltonian is characterized by the (d −1)th homotopy class of the transition function for real wavefunctions [9,50,51]. Then, we use the equivalence between theformalism in the smooth gauge and that in the real gauge:

πd [U (N )/O(N )]

GaugeDOF πd−1[O(N )], (8)

where d = 1, 2, which can be derived from the exact sequenceof homotopy groups (see Appendix A). We demonstrate therelation between the smooth gauge and the real gauge inmore detail for d = 1, 2 in the following Secs. III and IV,respectively.

Similarly, the 3D topological invariant corresponds to thethird homotopy class of G, and it is determined by the secondhomotopy class of G on C2zT -invariant planes. To show this,let us continuously deform the Brillouin torus T 3 as a 3-sphereS3 as shown in Figs. 2(a) and 2(b). This is valid as long as1D and 2D topological invariants are all trivial because thenthe noncontractible 1D and 2D cycles can be shrunk to apoint without changing the 3D topological invariant as longas the three-dimensional manifold itself is not shrunk to a

2We focus on d = 1, 2 here because we consider C2zT symme-try. However, this equation can be extended to higher-dimensionalsystems with PT symmetry with (PT )2 = 1. In general, it is validfor any d �= 4n for a positive integer n as shown in Appendix A.When d = 4n for some positive integer n, the band topology ischaracterized by the (2n)th Chern class, so the nontrivial bandtopology does not require PT symmetry and persists without thesymmetry. Therefore, the classification of the sewing matrix does notgive the full classification of band topology in the case.

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point. To make the 3-sphere in Fig. 2(b) from the 3-torusin Fig. 2(a), we deform each torus at a fixed kz ( �= ±π ) toa sphere, and that at kz = ±π to a point. Then, the kz = 0plane becomes the equator, and the kz = ±π planes becomethe north and south poles, respectively. Under C2zT , the wavefunctions on the northern hemisphere D3 are transformed tothe wave functions on the southern hemisphere and vice versa,while the equator S2 is invariant. Therefore, the effectivedomain consists of the upper hemisphere and its boundaryas shown in Fig. 2(b). The relevant homotopy group for theeffective domain is the relative homotopy group π3[M, X ],which classifies the maps D3 → M under the constraint∂D3 = S2 → X ⊂ M, where D3 is a three-dimensional diskand ∂D3 = S2 is the boundary of D3 [46–48,50]. In ourcase, M = U (N ), X = U (N )/O(N ), D3 is the upper hemi-sphere, and S2 is the equator as shown in Fig. 2(c). Therelative homotopy group π3[U (N ),U (N )/O(N )] has the formπ3[U (N )] × π2[U (N )/O(N )] as shown in Appendix A. Sincethe π3[U (N )] part comes from the gauge degrees of freedom,the third homotopy class is determined by the second homo-topy class on the invariant subspace, i.e., π2[U (N )/O(N )] π1[O(N )],

π3[U (N ),U (N )/O(N )]

π3[U (N )] π2[U (N )/O(N )]. (9)

We demonstrate this relation in Sec. VI.

III. FIRST HOMOTOPY CLASS

Here, we review the correspondence between the 1D wind-ing number of G in a smooth gauge and the first Stiefel-Whitney number w1 in a real gauge [6] since the same ideais used to derive the correspondence between the secondhomotopy class of G and the second Stiefel-Whitney numberin the next section.

Let us suppose that |unk〉 is smooth and the sewingmatrix G is defined in this basis. Then, we perform a gaugetransformation to get new basis states |unk〉 = Umn(k)|umk〉such that G(k) = U †(k)G(k)U ∗(k) and U (k) is smooth for0 < k < 2π , where 0 � k < 2π parametrizes a closed loopin the C2zT -invariant plane. If we require the reality conditionG(k) = 1 for the new basis, we have det[U †(k)G(k)U ∗(k)] =det G(k) = 1, so ∂k log det U (k) = 1

2∂k log det G(k). We havea transition function tmn ≡ 〈um0|un2π 〉 = U ∗

pm(0)Upn(0 + 2π )since 〈up0|uq2π 〉 = δpq due to the smoothness of the originalbasis. Its determinant is given by the winding number of G,which we write as w, namely, det t = det[U ∗(0)U (2π )] =exp[

∫ 2π

0 ∂k log det U (k)] = exp[ 12

∫ 2π

0 ∂k log det G(k)] =(−1)w. As the first Stiefel-Whitney number w1 is defined by(−1)w1 = det t , we have w1 = w modulo 2.

The above construction shows the relation (8). Here,det t = ±1 characterizes π0[O(N )] because t ∈ O(N ),and exp[

∫ 2π

0 ∂k log det U (k)] = ±1 characterizes thegauge-invariant part of π1[U (N )/O(N )]. Let us explainmore about this. Although U is not periodic when det t = −1because then det U is antiperiodic, U is periodic as anelement of U (N )/O(N ) [recall U (2π ) = U (0)t]. Smoothgauge transformations can change the winding number of U ,but it does not change the periodic condition of U . Therefore,among nontrivial elements in π1[U (N )/O(N )], only the

FIG. 3. Gauge transformation from a real to a smooth complexgauge in a C2zT -invariant plane. (a) C2zT -invariant 2D Brillouinzone covered by two patches A and B in a real gauge. (b) Thepatch A whose kx = π line is contracted to a point. (c) The gaugetransformation matrix U on the patch A.

loops along which det U changes sign is robust against gaugetransformations.

IV. SECOND HOMOTOPY CLASS

In this section, we show that the second homotopy classof G in a smooth gauge corresponds to the second Stiefel-Whitney number in a real gauge. Below, we begin with thedefinition of the second Stiefel-Whitney number in a realgauge, and then go to a smooth gauge. The gauge transfor-mation matrix is associated with the sewing matrix by Eq. (4).

We take a real gauge and cover the Brillouin zone toruswith two patches A and B, overlapping on the lines kx = 0 andπ [see Fig. 3(a)]. When the first Stiefel-Whitney numbers arenontrivial along both kx and ky directions, we should introducemore patches so that there exist discontinuous transitionsalong the ky direction [6]. However, we can always Dehn twistthe Brillouin zone in those cases as shown in Fig. 4 such thatonly one cycle has nontrivial w1 at most, and we take thenontrivial cycle to be along the kx direction [6]. We assumethat such a Dehn twist is done. Also, we take the transitionfunction at kx = π to be trivial. That is, we require that real oc-cupied states |unk〉 are smooth within the patches, but there canexist a nontrivial transition function on the equator defined by

tABmn (ky) ≡ ⟨

uAm(2π,ky )

∣∣uBn(0,ky )

⟩, (10)

which is an element of the orthogonal group O(N ) for Noccupied bands. The second Stiefel-Whitney number w2 isdefined by the 1D winding number of the transition functiontAB modulo 2.

FIG. 4. Dehn twist of the Brillouin zone. A Brillouin zone de-fined by 0 � kx, ky � 2π is Denn twisted to a tilted Brillouin zoneshown as a yellow (shaded) region. When both 1D cycles along kx

and ky have nontrivial 1D topological invariants, i.e., w1x = w1y = 1,we have a trivial 1D cycle k′

y by a Denn twist because w1y′ =w1x + w1y = 0 mod 2.

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Then, we consider a gauge transformation to smooth states|un(kx,ky )〉 via∣∣un(kx,ky )

⟩ = Umn(kx, ky )∣∣uA

m(kx,ky )

⟩, π � kx � 2π

(11)∣∣un(kx,ky )⟩ = Umn(kx, ky )

∣∣uBm(kx,ky )

⟩, 0 � kx � π

where U (kx, ky) is smooth for 0 � kx, ky � 2π . The gaugetransformation matrix U satisfies

tABmn (ky) = ⟨

uAm(2π,ky )

∣∣uBn(0,ky )

⟩= Ump(2π, ky)

⟨up(2π,ky )

∣∣uq(0,ky )⟩U ∗

nq(0, ky)

= Ump(2π, ky)δpqU ∗nq(0, ky), (12)

where we used that |unk〉 is smooth in the last line. Bychoosing a gauge U (0, ky) = 1, we have

U (2π, ky) = tAB(ky) ∈ O(N ). (13)

We further require that U (π, ky) is independent of ky, i.e.,

U (π, ky) = U0 ∈ U (N ), (14)

as shown in Fig. 3. It is possible to take this gauge because the1D topological invariant, the first Stiefel-Whitney number, istrivial along the ky direction.

Now, the information on the wave-function topology, en-coded in the transition function tAB in a real gauge, is reflectedin the unitary matrix U under the constraint of Eq. (13).Since U is constant on kx = 0 and π lines, the lines can beshrunk to a point as long as topology is concerned. Afterthe shrinking, the B region becomes a sphere, and the Aregion becomes a cap as shown in Fig. 3(b). All possibleU ’s are homotopically equivalent in the region B becausethey are classified by the homotopy group π2[U (N )] = 0.Therefore, we only need to study the homotopy class of Uon the region A. The homotopy group of U on the regionN with the boundary condition (14) is the relative homotopygroup π2[U (N ), O(N )] [see Fig. 3(c)]. Here, [U (N ), O(N )]means that U ∈ U (N ) inside the region A and U ∈ O(N ) onits boundary, which is the equator. Because π2[U (N )] = 0, therelative homotopy class of U is in one-to-one correspondencewith the homotopy class of U on its boundary, which isnothing but the homotopy class of the transition function tAB ∈π1[O(N )]. That is, π2[U (N ), O(N )] π1[O(N )]. Moreover,the relative homotopy group of U is isomorphic to the homo-topy group of G = UU T . In other words, π2[U (N ), O(N )] π2[U (N )/O(N )] [50], where the isomorphism is provided bythe projection from [U (N ), O(N )] to U (N )/O(N ). Therefore,

π2[U (N )/O(N )] π1[O(N )]. (15)

As the homotopy groups for smooth and periodic gaugetransfomations are trivial, the process described here providesan explicit mapping for the isomorphism in Eq. (8) in thed = 2 case.

Using this formulation of the second Stiefel-Whitney num-ber as a homotopy class of the sewing matrix, we can simplyderive the unique characteristic of the Stiefel-Whitney num-bers, the Whitney sum formula [6,51], if we require somenatural algebraic rules for the second homotopy classes on theBrillouin zone torus. Let us first consider a real gauge andsuppose that the occupied bands are grouped into blocks Bi

of bands isolated from each other, so that different blocks arenot connected by transition functions. For example, transitionfunctions are block diagonalized when there are finite-energygaps between blocks, though a gapped energy spectrum isnot necessary in general to have a block-diagonal form oftransition functions. On the Brillouin zone torus having non-contractible 1D cycles along kx and ky directions, the secondStiefel-Whitney number of the whole occupied bands ⊕Bi

is related to the Stiefel-Whitney numbers of blocks by theWhitney sum formula [6,51]

w2(⊕iBi ) =∑

i

w2(Bi ) +∑i �= j

wx1(Bi )w

y1(B j ), (16)

where wa=x,y1 is the first Stiefel-Whitney number along ka=x,y.

The appearance of the second term in the summation is aunique characteristic of the second Stiefel-Whitney number.

From the relation between the transition function in areal gauge and the sewing matrix in a smooth gauge derivedabove, we can infer that the Whitney sum formula should beapplicable to the blocks that decouple the sewing matrix ina smooth gauge. For instance, let us consider two blocks B1

and B2 of occupied bands that block diagonalize the sewingmatrix as

G(k) =(

eiθ1(k)G(0)1 (k) 0

0 eiθ2(k)G(0)2 (k)

), (17)

where the U (1) factor eiθi=1,2 of each block is singled out.Let N1 and N2 be the number of the bands in the blocks B1

and B2, respectively. Then, the second homotopy class of G :T 2 → U (N1)/O(N1) × U (N2)/O(N2) is determined by thesecond homotopy classes of G(0)

1 ∈ SU (N1)/SO(N1), G(0)2 ∈

SU (N2)/SO(N2), and (eiθ1 , eiθ2 ) ∈ U (1) × U (1) T 2. Theparities of the second homotopy class for G(0)

1 and G(0)2

correspond to w2(B1) and w2(B2), respectively. Because thegenerators of π2[SU (Ni )/SO(Ni )] for i = 1, 2 are mapped tothe generators of π2[U (N1 + N2)/O(N1 + N2)] by the inclu-sion maps, we have w2(B1 ⊕ B2) = w2(B1) + w2(B2) whenthe U (1) × U (1) part is neglected. For the map T 2 → U (1) ×U (1), we can define the degree of the map as a homotopyinvariant

1

(2π )2

∫BZ

d2k(∂kx θ1∂kyθ2 − ∂kx θ2∂kyθ1

)

= wx1(B1)wy

1(B2) − wx1(B2)wy

1(B1) mod 2, (18)

where we used that θi=1,2(k) is homotopically equivalent towx

1(Bi )kx + wy1(Bi )ky because they have the same 1D winding

number: wj1(Bi ) along k j=x,y. If we require that this homo-

topy invariant contributes to the two-dimensional topologicalinvariant, that is, the second Stiefel-Whitney invariant, weobtain the Whitney sum formula w2(B1 ⊕ B2) = w2(B1) +w2(B2) + wx

1(B1)wy1(B2) − wx

1(B2)wy1(B1). The generaliza-

tion to the cases with many blocks is straightforward.

V. STRONG TOPOLOGICAL INVARIANT IN 3D

Let us apply the results obtained above to the 3D topolog-ical insulator protected by C2zT symmetry. For this, we firstreview the definition of the 3D topological invariant in a real

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gauge and study the stability condition for the correspondingtopological phase.

Analogous to the Fu-Kane-Mele invariant, one can definea 3D strong topological invariant by using w2(0) and w2(π )defined on the kz = 0 and π planes, respectively, as

�3D ≡ w2(π ) − w2(0), (19)

which is identical to the Z2-invariant proposed in [7]. Be-cause w2(π ) = w2(0) in weakly coupled layered systems, thenonzero �3D is a 3D strong invariant [7]. In this respect, thephase with �3D = 1 can be called a 3D strong Stiefel-Whitneyinsulator (SWI). Below, we show that the 3D strong SWI is awell-defined stable topological phase only when all the Chernnumbers are trivial: cxy

1 = cyz1 = czx

1 = 0 where ci j1 indicates

the Chern number defined in the kik j plane. Here, a stabletopological phase indicates that its 3D topological invariantremains intact against adding atomic insulators [12], whose2D or 3D band topology is trivial (however, their 1D invariantsmight be nontrivial because 1D invariants are related to theWannier centers [5,52]).

To address the stability of the 3D strong SWI, let usconsider the Whitney sum formula which can be applied toC2zT -symmetric 2D BZ torus [6,51]. According to Eq. (16),w2 is fragile against adding bands with w2 = 0 if they havenontrivial w1 [6,17], although it is a stable K-theory invariant:it is stable against adding bands with w2 = 0 and w1 = 0. Forinstance, if a block B′ of bands is added to the original blockB, w2 changes as

δw2 = w2(B ⊕ B′) − w2(B)

= w2(B′) + wx1(B)wy

1(B′) + wx1(B′)wy

1(B) (20)

on both kz = 0 and π planes. Even when w2(B′) = 0, δw2

can be nonzero when wx,y1 (B′) is nontrivial unless wx

1(B) =w

y1(B) = 0. The corresponding change of �3D is given by

δ�3D = �3D(B ⊕ B′) − �3D(B)

= cxz1 (B)wy

1(B′) + cyz1 (B)wx

1(B′)(mod 2), (21)

where B′ is assumed to be from an atomic insulator, suchthat w

x,y1 (B′) are the same in both kz = 0 and π planes, and

�3D(B′) = 0. In order to define �3D independent of addingatomic insulators, we should require that cxz

1 (B) = cyz1 (B) =

0. Since cxy1 = 0 is always imposed by C2zT symmetry, we

conclude that �3D becomes a well-defined stable topologicalinvariant only when all the Chern numbers vanish in the BZ.

VI. THIRD HOMOTOPY CLASS

Let us now show that �3D is in fact equivalent to themagnetoelectric polarizability P3, defined by the effective La-grangian Ltop = P3E · B. First, we assume that all the Chernnumbers are trivial, i.e., cxy

1 = cyz1 = czx

1 = 0, and thus �3D isa stable topological invariant. Under this assumption, we cantake a smooth gauge over the whole Brillouin zone [18,53]. Ina smooth gauge, P3 takes the form of the 3D Chern-Simonsinvariant [18,54]

P3 = 1

8π2

∫BZ

d3k εi jkTr

[Ai∂ jAk − 2i

3AiAjAk

], (22)

where Amn(k) = 〈umk|i∇k|unk〉 is the Berry connection. SinceA∗

i (k) = G−1(k)(C−12z )i jA j (−C2zk)G(k) − G−1(k)i∇ki G(k)

in C2zT -symmetric systems, we have [43]

2P3 = 1

24π2

∫BZ

d3k εi jkTr[(G−1∂iG)(G−1∂ jG)(G−1∂kG)],

(23)

which is nothing but the 3D winding number of the sewingmatrix G. Let us note that, under the gauge transformation|unk〉 → |u′

nk〉 = Umn(k)|umk〉, 2P3 changes as

δ(2P3) = 2 × 1

24π2

∫BZ

Tr(U −1dU )3 ∈ 2Z. (24)

Therefore, 2P3 is a Z2 topological invariant well definedmodulo two.

Equation (9) implies that the 3D winding number of Gis determined by the second homotopy class of G on twoC2zT -invariant planes with kz = 0 and π , respectively. Let usprove this explicitly for the simplest case with two occupiedbands (N = 2) neglecting the U (1) factor. This assumption isgood enough to determine the topological invariant modulotwo because π3[U (N )] π3[SU (2)] and π2[U (N )/O(N )] π2[SU (2)/SO(2)] modulo two for all N � 2.

Let us note that SU (2) S3 and SU (2)/SO(2) S2,which can be obtained from the fact that a SU (2) el-ement U = a0 + ia1σ1 + ia2σ2 + ia3σ3 has four real co-efficients a0, . . . , a3 satisfying a2

0 + a21 + a2

2 + a23 = 1, and

SU (2)/SO(2) elements are the ones with a2 = 0. Then, thewinding number of G : T 3 → SU (2) S3 is determined bythe degree of G, which is given by the number of points inT 3 that is mapped to a given element u ∈ SU (2) [18]. Thedegree does not depend on the choice of u [18]. If we chooseu ∈ SU (2)/SO(2), when a point with kz �= 0 or π is mappedto u, the point has a partner related by C2zT that is mapped tothe same element u because u satisfies Eq. (2) and uT = u. Inthis case, a pair of points contributes an even number to thedegree of G. Accordingly, the parity of the degree is given bythe sum of the degree computed on the kz = 0 and π planes,i.e., the degree of the map G : T 2 → SU (2)/SO(2) S2 onthese planes. In other words, 2P3 = w2(π ) + w2(0) modulotwo because the degree of the map on the planes is identical tothe second Stiefel-Whitney number. Using w2 = −w2 modulotwo, we eventually obtain

�3D = 2P3 mod 2. (25)

VII. BULK-BOUNDARY CORRESPONDENCE

Given the relation in Eq. (25), the bulk-boundary corre-spondence of the 3D strong SWI can be described by usingthe known topological effective action [54]

S(bulk)top [A] = P3

16π

∫dt d3x εi jklFi jFkl , (26)

where Fi j = ∂iA j − ∂ jAi is the electromagnetic fieldstrength, and we take h = c = e = 1.

To study the boundary effect, let us consider a geometrywith a 3D strong SWI on one side with x < 0 and the vacuumon the other side with x > 0, which is modeled by P3(t, x) =P3(−x). After integration by parts, the effective action can

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be written as a boundary action

Stop[A] = 1

16π

∫R4

dt d3x P3(t, x)εi jkl∂i(4A j∂kAl ),

= P3

4π

∫x=0

dt d2x εi jkAi∂ jAk, (27)

which means that the bulk topological term induces thesurface quantum Hall effect with Hall conductivity σ

(surf)H =

P3/2π . In other words, the Chern number on the surface isgiven by

c(surf)1 = P3 mod 1 (28)

because σH = c1/2π . The surface state with c(surf)1 = 1

2 can berealized in two different ways depending on the symmetry ofthe system. Namely, it can be either a Chern insulator withhalf-quantized Hall conductance as in axion insulators [55] ora semimetal with an odd number of Dirac points as in time-reversal-invariant 3D topological insulators [56].

Here, we consider the case where C2z and T symmetries arebroken individually whereas the combined symmetry C2zT ispreserved. If both C2z and T are the symmetries of the system,the 3D strong SWI is not allowed when T 2 = 1 [7] becausew2(0) = w2(π ) as C2z eigenvalues indicate [6]. On the otherhand, when T 2 = −1, �3D = 1 is allowed, but it is identicalto the well known Fu-Kane-Mele invariant [7] since the bulkP3 is nontrivial [18].

In a 3D strong SWI, both gapless and gapped states appearon the surface as shown in Fig. 1(b). To understand this, let usconsider an orthorhombic geometry. On the top and bottomsurfaces, which are C2zT invariant, insulating states are notallowed because C2zT symmetry requires the vanishing ofthe Chern number. Instead, there appears an odd number ofDirac points, whose π Berry phase is protected due to thequantization of the Berry phase by C2zT [7]. On the otherhand, side surfaces are gapped because C2zT symmetry isbroken and thus 2D Dirac points cannot be protected. So,the side surfaces become Chern insulators with half-quantizedHall conductance with c1 = n ± 1

2 where n is an integer. Thesign of c1 on the side surfaces is related by C2zT symmetrythrough

c1(x) = −c1(C2zx), (29)

where c1 = (1/2π )∫

BZ d2k TrF · n, and n is the surface nor-mal unit vector pointing outward, and F = dA − iA × A isthe Berry curvature. It means that the front side surface andthe back side surface form two domains with different Chernnumbers. Therefore, chiral 1D states appear on side hingesthat are the boundaries of the two different domains. Let usnote that the stability condition for the 3D strong SWI, thatis, the vanishing of the bulk Chern numbers, prohibits otheranomalous surface states on the side surfaces, and thus thechiral hinge states can become well localized.

VIII. TIGHT-BINDING MODEL

To illustrate the higher-order band topology of a 3D strongSWI, let us perform a tight-binding model analysis. We startwith a Hamiltonian describing a 3D Dirac semimetal withboth PT and C2zT symmetries, where P indicates spatial

inversion

HDSM = sin kx1 + sin ky2

+ (−2 + cos kx + cos ky + cos kz )3, (30)

where 1 = σx, 2 = τyσy, and 3 = σz. The symmetryrepresentations for PT and C2zT are given by PT = Kand C2zT = K . Two Dirac points appear at (kx, ky, kz ) =(0, 0,±π/2), respectively, each of which is protected by PTsymmetry. Each Dirac point carries a Z2 monopole charge[8–10], which is identical to the nontrivial w2 on a closedmanifold wrapping the Dirac point. Since the energy gap isfinite except at (kx, ky, kz ) = (0, 0,±π/2), the sphere wrap-ping a Dirac point can be deformed to two parallel planes withkz = 0 and π , respectively. Then, the monopole charge of aDirac point is given by the difference of w2(0) and w2(π ).

Adding C2zT -preserving perturbations that open the bulkand surface band gaps, we have

H = HDSM + v sin kz4 + m1414 + m2424, (31)

where 4 = τxσy, 14 = τxσz, and 24 = −τz. v �= 0 opensthe bulk gap because it breaks PT symmetry whereas PT -preserving terms m14 �= 0 and m24 �= 0 open the gap onthe side surfaces (a closely related model was constructedin [22]). Since m14 and m24 deform each Dirac point to amonopole nodal line in the bulk Brillouin zone, H can be re-garded as a monopole nodal line semimetal with an additionalPT -breaking parameter v that transforms as the in-planemagnetic field. As long as the perturbations are small such thatthe band gap does not close on kz = 0 and π planes, w2(π ) −w2(0) = 1 mod 2 should be maintained. Wilson loop calcula-tions in Figs. 5(a) and 5(b) show that w2(0) = 1 and w2(π ) =0 because w2 is given by the number of linear crossing pointsat = π modulo two, where is the phase of the eigenvalueof the Wilson loop operator [6,9,10,14,16,49,57,58] (see alsoAppendix B). Our finite-size calculations in Fig. 5(c) with14 × 14 × 14 unit cells show that the system has anomalousin-gap states. The hinge and surface states coexist as shown inFigs. 5(d)–5(g), which are from the first four highest-energyoccupied states below the Fermi level at half-filling. Thelinearly dispersing spectrum of the in-gap states is visible ifwe calculate the band structure with partial open boundaryconditions as shown in Figs. 5(h) and 5(i).

IX. DISCUSSION

We have shown that the second Stiefel-Whitney number isa homotopy invariant that determines the second homotopyclass of the sewing matrix for C2zT . Taking into account therelated results for d = 1 and 3 reported before, we concludethat the d-dimensional topological invariant is the measureof the dth homotopy class of the sewing matrix in C2zT -symmetric systems.

Since the homotopy equivalence of the matrix represen-tation of symmetry groups classifies topological crystallineinsulators in principle [20], we expect that all known topo-logical crystalline insulators can be expressed as a homotopyinvariant of some sewing matrix. For instance, the Fu-Kaneinvariant [59] is indeed defined as the homotopy invariantof the sewing matrix. The mirror Chern number can also be

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JUNYEONG AHN AND BOHM-JUNG YANG PHYSICAL REVIEW B 99, 235125 (2019)

FIG. 5. Numerical calculation using the model in Eq. (31). v =0.5, m14 = 0.5, and m24 = 0.3. (a), (b) Wilson loop calculation on(a) kz = 0 and (b) kz = π planes. (c)–(g) Finite-size calculation with14 × 14 × 14 unit cells. (c) Energy eigenvalues in an increasingorder. Near the half-filling (n = 2 × 143 = 5488), anomalous statesappear within the bulk gap |Eg| ≈ 0.37. (d)–(g) Density profile ofin-gap states computed by using the (d) first, (e) second (f), third (g),and fouth highest occupied states below Fermi level at half-filling.(h) Band structure of a model which is periodic in the (x, y) direction,and has 20 unit cells in the z direction. The Dirac point in the bulkband gap originates from the top and bottom surfaces. (i) Bandstructure of the model which has 20 × 20 unit cells in the (x, y)direction but periodic in the z direction. Hinge states appear withinthe bulk band gap.

interpreted as the second homotopy class of the sewing matrix(see Appendix C). Another example is the magnetoelectricpolarizability. In general, the magnetoelectric polarizabilityis quantized in the presence of a symmetry operation thatreverses the space-time orientation regardless of whether it issymmorphic or nonsymmorphic [60]. When the magnetoelec-tric polarizability is quantized by a symmorphic symmetry,it is known to be expressed by the 3D winding number ofthe sewing matrix [43]. We further show in Appendix D thatthe same is true for nonsymmorphic symmetries. Namely, thetopological invariants protected by a nonsymmorphic symme-try can also be described as the obstruction to a momentum-independent representation of the relevant sewing matrix.Extending the analysis to include other magnetic space groupsis definitely one interesting direction for future research.

Note added. Recently, we have found a related paper [61]that also identifies the presence of chiral hinge states in C2zT -protected insulators with P3 = 1.

ACKNOWLEDGMENTS

We thank Benjamin J. Wieder, Bogdan Andrei Bernevig,and Yoonseok Hwang for helpful comments to our

manuscript. J.A. was supported by Grant No. IBS-R009-D1.B.-J.Y. was supported by the Institute for Basic Science inKorea (Grant No. IBS-R009-D1) and Basic Science ResearchProgram through the National Research Foundation of Korea(NRF) (Grant No. 0426-20190008), and the POSCO ScienceFellowship of POSCO TJ Park Foundation (Grant No. 0426-20180002). This work was supported in part by the US ArmyResearch Office under Grant No. W911NF-18-1-0137.

APPENDIX A: SOME PROPERTIESOF HOMOTOPY GROUPS

In this Appendix, we prove some properties of homotopygroups we use in the main text. The main tool to be used is thelong exact sequence of homotopy groups [46,48,50]:

· · · ∂p+1−−→ πp(X )i∗p−→ πp(M )

j∗p−→ πp(M, X )

∂p−→ πp−1(X )i∗p−→ · · · , (A1)

where ip : X → M and jp : M → (M, X ) are inclusions, i∗pand j∗pare maps for homotopy groups induced by ip and jp,and ∂ is the restriction to the boundary. This sequence is exactbecause the image of a map is the kernel of the next map, e.g.,im i∗p = ker j∗p. It is also valid when πp(M, X ) is substituted byπp(M/X ) because the two homotopy groups are isomorphic[48,50]:

· · · ∂p+1−−→ πp(X )i∗p−→ πp(M )

j∗p−→ πp(M/X )

∂p−→ πp−1(X )i∗p−→ · · · . (A2)

1. Equivalence between real and smooth gauges

Let us prove Eq. (8), that is, πd [U (N )/O(N )]/im j∗d πd−1[O(N )] when d �= 4n for a positive integer n. It can beproved for arbitrary N when d = 1, 2, which are dimensionsstudied in the main text, whereas we need the large-N limit ingeneral dimensions, This follows from the exact sequence inEq. (A2). In our case, M = U (N ), and X = O(N ). We have

· · · → πd [U (N )]j∗d−→ πd [U (N )/O(N )]

∂d−→ πd−1[O(N )]i∗d−1−−→ πd−1[U (N )] → · · · . (A3)

Then, we have

πd [U (N )/O(N )]

im j∗d ker i∗d−1, (A4)

where we used the exactness of maps ker ∂d = im j∗dand im ∂d = ker i∗d−1 and the group isomorphism theoremπd [U (N )/O(N )]/ ker ∂d im ∂d . Notice that i∗d−1 is a trivialmap for d �= 4n for a positive integer n when N is largeenough. When d is odd, it is because πd−1[U (N )] = 0 for d �2N . In particular, π0[U (N )] = π2[U (N )] = 0 for all N . Whend = 2, i∗d−1 is trivial because orthogonal group elements havequantized determinants +1 or −1, so that they cannot havea winding of the determinant (recall that π1[U (N )] is char-acterized by the winding number of the determinant of theunitary matrix). When d = 6, the map i∗5 is trivial becauseπ5[O(N )] = 0. Bott periodicity then shows that the same is

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true for 2 + 8m and 6 + 8m dimensions for a positive integerm when N is large enough, i.e., d � 2N and d � N − 1. Onthe other hand, when d = 4n for a positive integer n, i∗d−1 isnot trivial. This is related to the fact that the reality conditionon wave functions (equivalently, PT symmetry) does notrequire that the (2n)th Chern class vanishes, and the Chernclass in a real gauge is called the Pontrjagin class [62]. Let usrecall that the (2n)th Chern number is given by the (4n − 1)thnontrivial homotopy of the transition function. The (2n)thChern class of real wave functions does not vanish becausethe nontrivial homotopy class of the transition function inπd−1[O(N )] survives as an element in πd−1[U (N )]. Accord-ingly,

ker i∗d−1 πd−1[O(N )] for d /∈ 4Z+, (A5)

where Z+ is the set of positive integers. This finishes theproof.

2. Third homotopy group of C2zT

Let us prove that

π3

[U (N ),

U (N )

O(N )

]= π3[U (N )] × π2

[U (N )

O(N )

]. (A6)

This follows from the long exact sequence

· · · ∂∗4−→ π3

[U (N )

O(N )

]i∗3−→ π3[U (N )]

j∗3−→ π3

[U (N ),

U (N )

O(N )

]

∂3−→ π2

[U (N )

O(N )

]i∗2−→ π2[U (N )]

j∗2−→ · · · (A7)

and that the map Im i∗n = 0 for n = 2, 3. Im i∗2 = 0 becauseπ2[U (N )] = 0, and Im i∗3 = 0 because Tr(U −1dU )3 = 0 forU ∈ U (N )/O(N ).

APPENDIX B: WILSON LOOP METHOD

In this Appendix, we show the connection between thesecond homotopy class of the sewing matrix for C2zT and thewinding number of the Wilson loop spectrum in an invariantplane. This provides a new insight into the Wilson loopmethod [6,9,10,14,16,49,57,58].

We first define a Wilson line operator for the occupiedstates on the line connecting k and k′ by

Wk→k′ = limδ→0

Fk′−δFk′−2δ . . . Fk+δFk, (B1)

where

(Fk )mn = 〈umk+δ|unk〉 , (B2)

and m, n are indices for occupied states. The transition matrixF satisfies the following equation in C2zT -symmetric systems:

(F ∗k )mn = 〈umk+δ|unk〉∗ = 〈C2zTumk+δ|C2zTunk〉

= G∗pm(k + δ) 〈upk+δ|uqk〉 Gqn(k)

= [G†(k + δ)FkG(k)]mn. (B3)

It follows that

W ∗k→k′ = lim

δ→0F ∗

k′−δF∗

k′−2δ . . . F ∗k+δF

∗k

= G†(k′)Wk→k′G(k). (B4)

FIG. 6. Wilson line operator in a C2zT -invariant Brillouin zone.(a) W (kx, ky ) is the Wilson line operator W(kx ,−π )→(kx ,ky ). (b) Deforma-tion of (a) after kx = −π, kx = π , and ky = −π lines are contractedto a point.

Therefore, we find that

G(k′) = Wk→k′G(k)W Tk→k′ . (B5)

For simplicity, we assume that all 1D topological invariantsare trivial. Then, we can take a gauge G(kx,−π ) = 1 such that

G(kx, ky) = W(kx,−π )→(kx,ky )WT

(kx,−π )→(kx,ky ). (B6)

Because we are in a smooth gauge, we have

1 = G(kx,−π )

= G(kx, π )

= W(kx,−π )→(kx,π )WT

(kx,−π )→(kx,π ), (B7)

so the Wilson loop operator belongs to the orthogonal groupat ky = π :

W(kx,−π )→(kx,π ) ∈ SO(N ). (B8)

It belongs to SO(N ) ⊂ O(N ) because it is continuously con-nected to the identity element W(kx,−π )→(kx,−π ) = 1.

Let us contract the kx = −π, kx = π , and ky = −π linesto a point, which is possible due to the assumption thatthe 1D topology is trivial, as shown in Figs. 6(a) and 6(b).As we show in Appendix A, the relative homotopy classof π2[U (N ), SO(N )] of W (kx, ky) ≡ W(kx,−π )→(kx,ky ) is deter-mined by its homotopy class on the boundary π1[SO(N )].Notice that the relative homotopy class of W is one-to-onecorrespondence with the second homotopy class of G as de-rived in Appendix A. Also, the homotopy class in π1[SO(N )]is given by the winding number of the Wilson loop opera-tor W [kx] ≡ W(kx,−π )→(kx,π ). Therefore, we conclude that thesecond homotopy class of G(kx, ky) is in one-to-one corre-spondence with the 1D winding number of the Wilson loopoperator W [kx]. In practice, one obtains the winding numberof the Wilson loop operator from the winding pattern of itsspectrum, which can be calculated in a gauge-invariant way.

APPENDIX C: MIRROR CHERN NUMBER

Here, we show that the mirror Chern number can be repre-sented as the second homotopy class of the sewing matrix forthe mirror operator M. We follow the same procedure we usedin the C2zT -symmetric case.

Consider the sewing matrix for the z-mirror M operator ina two-dimensional system, where k = (kx, ky)

Gmn(k) = 〈ψnMk|M|ψnk〉 . (C1)

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As mirror operator satisfies M2 = 1 (when M2 = −1, we canconsider M ′ = iM that satisfies M ′2 = 1),

G†(k) = G(Mk). (C2)

The sewing matrix transforms by

Gmn(k) → G′mn(k) = [U †(Mk)G(k)U (k)]mn, (C3)

under the gauge transformation|unk〉 → |u′nk〉 = Umn(k)|umk〉,

where G′mn(k) = 〈u′

mMk|M|u′nk〉.

In a mirror-invariant plane where Mk = k, G is a unitaryHermitian matrix, so it has the following form:

G(k) = U †(k)

(1N×N 0

0 −1M×M

)U (k), (C4)

where U (k) ∈ U (N + M ) is a gauge transformation neededto diagonalize the sewing matrix G(k). Since diagonal U ∈U (N ) × U (M ) does not change the matrix G, the sewingmatrix belongs to the quotient space called the complexGrassmannian manifold

G(k) ∈ U (N + M )/U (N ) × U (M ). (C5)

It can have a second homotopy class because πn[U (N +M )/U (N ) × U (M )] = 0 for n = odd and Z for n = even.

Following the same logic used for C2zT symmetry, we canderive the relation between the mirror Chern number and thewinding number of G. Let us consider the same sphericalgeometry used for the C2zT -symmetric case [see Fig. 3(a)],and start from the mirror eigenstate basis |un(kx,ky )〉 with G =diag(1N×N ,−1M×M ). In the eigenstate basis, the Brillouinzone is covered with two patches, A and B, overlapping onkx = π and 2π . We take a gauge where the transition functionis trivial on kx = π . Then, the transition matrix tAB(φ) =〈uA

n(2π,ky )|uBn(0,ky )〉 takes the form

tAB =(

tAB+ 00 tAB

−

), tAB

+ ∈ U (N ), tAB− ∈ U (M ). (C6)

The winding number of tAB+ and tAB

− gives the Chern numberof the sector with mirror eigenvalues +1 and −1, respectively.

After we transform to a smooth gauge, the winding numberof the transition function is encoded in the second homotopyclass of the sewing matrix G. Here, we assume that the totalChern number vanishes in order to take a smooth gauge atthe cost of giving up a uniform sewing matrix. We first showthat the relative homotopy class of the gauge transformationmatrix U needed to go to a smooth gauge corresponds to thehomotopy class of the transition function. Then, we get thedesired result by Eq. (C4).

Let us first show that U satisfies the constraint that it equalsto the transition matrix on the equator. We consider a gaugetransformation from mirror eigenstates |un(kx,ky )〉 to smooth

states |un(kx,ky )〉 defined by |uA/Bn(kx,ky )〉 = Umn(kx, ky)|um(kx,ky )〉.

The gauge transformation matrix U satisfies tABmn (ky) =

U ∗pm(2π, ky)Upn(0, ky), where we used that |unk〉 is smooth

such that 〈up(kx,ky )|uq(kx,ky )〉 = δpq. By choosing the gaugeU (0, ky) = 1, we have

U (2π, ky) = tAB(ky) ∈ U (N ) × U (M ), (C7)

on the equator.

Next, we show that the relative homotopy class of Uis given by the homotopy class of the transition functiontAB. This follows from the exact sequence in Eq. (A1).In our case, M = U (N + M ), and X = U (N ) × U (M ). Asπ2[U (N + M )] = 0, we have

0j∗2−→ π2[U (N + M ),U (N ) × U (M )]

∂2−→ π1[U (N ) × U (M )]i∗2−→ π1[U (N + M )]

j∗1−→ · · · , (C8)

where 0 = {1}. Then, π2[U (N + M ),U (N ) × U (M )] ker i∗2 because ker ∂2 = im j∗2 = 1 and im ∂2 = ker i∗2. Noticethat ker i∗2 is composed of elements whose total windingnumber vanishes, i.e., the total Chern number is trivial, suchthat the nontrivial element in the group characterizes themirror Chern number. We can also show that the homotopygroup for G is in one-to-one correspondence with the relativehomotopy group of U in the same way as we did for C2zTsymmetry. In conclusion, the second homotopy class of Gcorresponds to the mirror Chern number.

APPENDIX D: QUANTIZATION OFMAGNETOELECTRIC POLARIZABILITY

In a system symmetric under the space-time-orientation-reversing transformation g, regardless of whether it is sym-morphic or nonsymmorphic, the magnetoelectric polarizabil-ity is quantized [60]. Here, we show that the magnetoelectricpolarizability is given by the winding number of the sewingmatrix of g.

Let us consider a system that is symmetric under a space-time transformation g : (r, t ) → (Or + t, sgt ), where O is apoint group element. Then, the symmetry operator Ug acts onthe position operator r and the pure imaginary number i as

U −1g rUg = Or + t,

(D1)U −1

g iUg = sgi.

The symmetry representation for the Bloch states is given by

Gmn(k) = ⟨ψmsgOk

∣∣Ug|ψnk〉 . (D2)

Accordingly, the cell-periodic part transforms by⟨umsgOk

∣∣Ug|unk〉 = ⟨ψmsgOk

∣∣eisgOk·rUge−ik·r|ψnk〉= ⟨

ψmsgOk∣∣eisgOk·rUge−ik·rU −1

g Ug|ψnk〉= ⟨

ψmsgOk∣∣eisgOk·re−isgk·O−1(r−t)Ug|ψnk〉

= eisgOk·t ⟨ψmsgOk

∣∣Ug|ψnk〉= eisgOk·tGmn(k) (D3)

such that

|unk〉 = eiOk·tGsg

mn(k)U −1g

∣∣umsgOk⟩, (D4)

where we used the notation introduced in [21]: fsg = f for

sg = 1 and fsg = f ∗ for sg = −1. Using this, one can show

that the Berry connection satisfies the following symmetry

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SYMMETRY REPRESENTATION APPROACH TO … PHYSICAL REVIEW B 99, 235125 (2019)

constraint:

Amn(k) = 〈umk|i∇k|unk〉= ⟨

U −1g up,sgOk

∣∣e−iOk·tG−sg

pm (k)i∇keiOk·tGsg

qn(k)∣∣U −1

g uq,sgOk⟩

= G−sg

pm (k)⟨U −1

g up,sgOk∣∣i∇k

∣∣U −1g uq,sgOk

⟩G

sg

qn(k) − δmnO−1t + G−sg

pm (k)i∇kGsg

pn(k)

= sgG∗pm(k)

⟨up,sgOk

∣∣i∇k∣∣uq,sgOk

⟩Gqn(k) + sgG∗

pm(k)i∇kGpn(k) − δmnO−1tsg

= G∗pm(k)O−1 · ⟨

up,sgOk∣∣i∇sgOk

∣∣uq,sgOk⟩Gqn(k) + sgG∗

pm(k)i∇kGpn(k) − δmnO−1tsg

= [sg(G−1(k)sgO−1 · A(sgOk)G(k) + G−1(k)i∇kG(k)) − O−1t]mnsg

≡ [sg(G−1(k)A(k)G(k) + G−1(k)i∇kG(k)) − O−1t]mn

sg

≡ [sgAG(k) − O−1t]mn

sg

, (D5)

where we introduced two notations A(k) = sgO−1 · A(sgOk) and AG(k) = G−1(k)A(k)G(k) + G−1(k)∇kG(k). The magneto-electric polarizability P3 then satisfies

P3 = P3sg

= 1

8π2

∫BZ

d3k εi jkTr

[A

sg

i ∂ jAsg

k + 2sgi

3A

sg

i Asg

j Asg

k

]

= 1

8π2

∫BZ

d3k εi jkTr

[(sgAG − O−1t)i∂ j (sgAG − O−1t)k + 2sgi

3(sgAG − O−1t)i(sgAG − O−1t) j (sgAG − O−1t)k

]

= 1

8π2

∫BZ

d3k εi jkTr

[AG

i ∂ j AGk − 2i

3AG

i AGj AG

k

]− 1

8π2

∫BZ

d3k εi jksg(O−1t)iTr[∂ j A

Gk

]

= 1

8π2

∫BZ

d3k εi jkTr

[Ai∂ j Ak − 2i

3AiA j Ak

]+ 1

24π2

∫BZ

d3k εi jkTr[(G−1∂iG)(G−1∂ jG)(G−1∂kG)]

= 1

8π2

∫BZ

d3k εi jk (sgO−1)ia(sgO−1) jb(sgO−1)kcTr

[Aa∂bAc(sgOk) − 2i

3AaAbAc(sgOk)

]+ 1

24π2

∫Tr(G−1dG)3

= 1

8π2

∫BZ

d3k εabc det(sgO−1)Tr

[Aa∂bAc(sgOk) − 2i

3AaAbAc(sgOk)

]+ 1

24π2

∫Tr(G−1dG)3

= 1

8π2

∫BZ

d3k εabcsg det O−1Tr

[Aa∂bAc − 2i

3AaAbAc

]+ 1

24π2

∫Tr(G−1dG)3

= sg det O−1

8π2

∫BZ

d3k εabcTr

[Aa∂bAc − 2i

3AaAbAc

]+ 1

24π2

∫Tr(G−1dG)3

= (sg det O−1)P3 + 1

24π2

∫BZ

d3k εi jkTr[(G−1∂iG)(G−1∂ jG)(G−1∂kG)], (D6)

where we assumed that all the first Chern numbers are triv-ial to remove the term

∫d3k εi jk (O−1t)iTrFjk and the total

derivative term∫

d3k εi jkTr[∂i(G−1∂ jGAk )] [62] in the fifthline. We have obtained that

2P3 = 1

24π2

∫BZ

Tr(G−1dG)3 ∈ Z (D7)

for the symmetry operation with sg det O−1 = −1.

APPENDIX E: ANOMALOUS BOUNDARY STATESOF AXION INSULATORS

Let us comment on the general bulk-boundary corre-spondence of insulators with quantized magnetoelectric po-larizability, the so-called axion insulators. As shown inAppendix D, the magnetoelectric polarizability is quantized

by a space-time-orientation-reversing symmetry in general.Let g be orientation reversing. Then, on the surface,

c1(x) = −c1(gx) (E1)

because c1 = (1/2π )∫

BZ d2k TrF · n, and F · n changes signby operations that reverse the space-time orientation. Here,n is the surface normal unit vector pointing outward, andF = dA − iA × A is the Berry curvature. Accordingly, x andgx belong to different surface domains with opposite signsof Chern numbers if they are gapped. Using this, we cangenerate the real-space configuration of boundary states of ax-ion insulators protected by space-time-orientation-reversingsymmetries. Figure 7 shows anomalous boundary states ofaxion insulators [37–39,41–45].

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JUNYEONG AHN AND BOHM-JUNG YANG PHYSICAL REVIEW B 99, 235125 (2019)

FIG. 7. Anomalous boundary states of axion insulators. (a)–(g)Symmorphic symmetries: (a) P, (b) C2zT , (c) C4zT , (d) C6zT , (e) M =C2P, (f) C3P, (g) C4P, (h) C6P. (i)–(l) Nonsymmorphic symmetries:(i) glide M, (j) screw C2zT , (k) screw C4zT , (l) screw C6zT . Shadedgray regions are the (glide) mirror-invariant planes.

APPENDIX F: SURFACE STATES OF DIRAC SEMIMETALS

In this Appendix, we identify the bulk-symmetry-preserving terms that can gap out the surface states of aPT -symmetric spinless Dirac semimetal.

First, consider the effective Hamiltonian for a Diracsemimetal with two Dirac points at (0, 0, kz ) = (0, 0,±k∗):

Hkz (kx, ky) = kx1 + ky2 + (k2∗ − k2)3, (F1)

where P = σx, T = τxK , and

1 = σy, 2 = τzσz, 3 = σx. (F2)

At a fixed kz, the Hamiltonian describes a 2D Stiefel-Whitneyinsulator (normal insulator) when m0 > 0 (m0 < 0) if wedefine m0 = k2

∗ − k2z .

We now investigate the edge states of the Stiefel-Whitneyinsulator by considering a system occupying only a half-spacex > 0, following Ref. [63]. As the x direction is not periodic,we write the Hamiltonian in real space for the direction

H = (−i∂x )1 + ky2 + m(x)3

=

⎛⎜⎝

ky m − ∂x 0 0m + ∂x −ky 0 0

0 0 −ky m − ∂x

0 0 m + ∂x ky

⎞⎟⎠, (F3)

where m(x � 0) = m0, m(x � 0) = −m0, m(x) changessign at x = 0, and m0 > 0.

As the Hamiltonian is block diagonal, we first solve theSchrödinger equation Hu = Eu for the upper block using u =(u1, u2, 0, 0)T :

(m − ∂x )u2 = (E − ky)u1,(F4)

(m + ∂x )u1 = (E + ky)u2.

Applying u∗2(m − ∂x ) to the upper and u∗

1(m + ∂x ) to the lowerequation, we get

|(m + ∂x )u1|2 = (E2 − k2

y

)|u1|2,(F5)

|(m − ∂x )u2|2 = (E2 − k2

y

)|u2|2,using the anti-Hermiticity of ∂x which follows from the Her-miticity of H . From this we find |E | � |ky|.

We seek solutions satisfying the lowest bound E = ±ky

because we are interested in the in-gap states that are theclosest to the Fermi level. For E = ±ky, Eqs. (F4) and (F5)become

(E − ky)u1 = (m − ∂x )u2 = 0,(F6)

(E + ky)u2 = (m + ∂x )u1 = 0.

We have u2 = 0 and (m + ∂x )u1 = 0 when E = ky, and u1 =0 and (m − ∂x )u2 = 0 when E = −ky. Therefore, the edgestate, which exponentially decays into the bulk, is

u ∝ exp

(−

∫ x

ds m(s)

)(1, 0, 0, 0)T , (F7)

and its energy eigenvalue is E = ky. We can do the same forthe lower block to have

u ∝ exp

(−

∫ x

ds m(s)

)(0, 0, 1, 0)T , (F8)

and its energy eigenvalue is E = −ky. Thus, we have two edgestates of opposite chirality:

H edgekz

=(

ky 00 −ky

). (F9)

As they exist for every kz such that |kz| < k∗, these edge statesform double Fermi arcs on the surface x = 0 of the Diracsemimetal.

Next, we include other terms as perturbations. Define

4 = τxσz, 5 = τyσz, i j = [i, j]

2i. (F10)

Then, i j terms with i = 1, 2, 3 and j = 4, 5 are PT symmet-ric. Projecting the terms to the edge states

u+ = (1, 0, 0, 0)T , u− = (0, 0, 0, 1)T , (F11)

we find

edge14 = 0,

edge15 = 0,

edge24 =

(0 −ii 0

), (F12)

edge25 =

(0 11 0

),

edge34 = 0,

edge35 = 0.

24 and 25 serve as mass terms of H edge.Similarly, we find that 14 and 15 serve as mass terms if

we take the y direction finite. In conclusion, all the surfacesnormal to x or y are gapped if we include, e.g., 14 and 24

mass terms as is done in the main text.

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